Particles converted but materials are linked
A quick workaround would be to utilize the Random output of a Object Info node in the shader graph, plugging it into a Hue/Saturation node input. In this way you could randomly vary the hue or the saturation or both. I personally utilize a Bright/Contrast node in between to get the hues I need.

But I think the real issues is to be found in where exactly this material information is stored. You can conveniently decide weather to store it in mesh data or in object data opting in this drop-down menu in the material property panel.

If materials are assigned in mesh data (&quot;Data&quot;) every instance (meaning linked copy) of that object will have the those materials and if you change them they will change over all objects. Otherwise if materials are stored in the object data-block different linked object will be able to have different materials.
If you prefer proceeding this way you'll have to change this property before converting the particle system, in the original sphere.